By A. Lahiri

This quantity is an introductory textual content the place the subject material has been provided lucidly as a way to support self examine by way of the newcomers. New definitions are by means of compatible illustrations and the proofs of the theorems are simply obtainable to the readers. enough variety of examples were integrated to facilitate transparent knowing of the strategies. The booklet begins with the fundamental notions of class, functors and homotopy of continuing mappings together with relative homotopy. basic teams of circles and torus were taken care of besides the basic staff of masking areas. Simplexes and complexes are offered intimately and homology theories-simplicial homology and singular homology were thought of besides calculations of a few homology teams. The e-book can be best suited to senior graduate and postgraduate scholars of varied universities and institutes.

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**Extra resources for A First Course in Algebraic Topology**

**Sample text**

Then 0 i11 an open neighbourhood of i with the·property that p I 0 is a homeomorphism of 0 onto U. This shows that p is a local homeomorphism. However a local homeomorphism need not be a covering map as the following example shows. 7. Let Pt : (0, 3 )4 st be the restrictioQ of the map p : R 4 st defined by p(t) = e 2nit to the open interval (0, 3). Since p is a cov~ring map (see Ex. 2 and so its restriction Pt to the open set (0, 3) is a local homeomorphism. Pt is also a surjection. But since the complex number 1 e st has no neighbourhood evenly covered by Pt> Pt is not a covering map.

G = g Proof. Existence off* gives thatf(l) g(l). Since f * is closed, we also have f(O) g(O). 11. So,f- g. 11. This gives that f* null path. g is homotopic to a EXERCISES I. In R2 let A = ((x, y) : x = 0, - 1 Sy S I} and B = ((x, y) : 0 < x S l, y =cos nix}. Show that F = A u B is connected but not path connected. 32 ALGEBRAIC TOPOLOGY 2. In R2 let A = {(x, y) : 0 ~ x ~ l, y = xln, n e N} and B= {

5. Paths _ _ _ _ _ _ __ 1. Definitions Let X be a topological space and C denotes the unit interval [O, 1]. A path in Xis a continuous mapping/: C ~ X. The points/(O) and/(1) are called respectively the initial point and the terminal point of the path. It should be noted that the path is the mapping f and not the image /([0, 1]). If x is a fixed element of X, we shall denote by Ex the constant mapping Ex: [0, l] ~ X = given by Ex(t) x for every t e C. The path Ex is called a null path. 1. Ex is a path.